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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Algebraic Preliminaries

Algebraic Preliminaries

Chapter:
(p.1) 1 ALGEBRAIC PRELIMINARIES
Source:
Quantum Field Theory and Critical Phenomena
Author(s):

JEAN ZINN-JUSTIN

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198509233.003.0001

This chapter reviews algebraic identities about gaussian integrals, in particular Wick's theorem, a result also relevant for gaussian probability distributions. It discusses the steepest descent method, which reduces a certain type of integrals to gaussian expectation values. It then defines and discusses a few properties of dierentiation and integration in a Grassmann, that is, antisymmetric algebra, relevant for theories with fermion particles. In particular, gaussian integrals are calculated and general integrals are again reduced to gaussian expectation values. The chapter also recalls the concept of Legendre transformation, generating functional, functional dierentiation and the algebraic defiition of the determinant of an operator.

Keywords:   gaussian integrals, steepest descent method, Grassmann algebra, antisymmetric algebra, Legendre transformation

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